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g
MODELING DECISIONS OF CHOICE
AMONG FINITE ALTERNATIVES: APPLICATIONS TO
MARKETING AND TO TRANSPORTATION DEMAND THEORY
by
John R. Hauser
OR 038-74
October
1974
Supported in part by the U.S. Army Research Office
(Durham) under Contract No. DAHC04-73-C-0032
ABSTRACT
A NORMATIVE METHODOLOGY FOR MODELING INDIVIDUAL RESPONSE TO INNOVATION
by
John Hauser and Glen Urban
A methodology to improve the effectiveness of the design of innovation
is proposed based on knowledge in the fields of psychometrics, utility
theory and stochastic choice modeling. It is comprised of a consumer
response and a managerial design process. The design process is one of idea
generation, evaluation, and refinement while the consumer response is based
on consumer measurement, an individual choice model, and aggregation of
the individual choices. The consumer model interacts with the design process
by providing diagnostics on consumer perceptions, preference, choice, and
segmentation, as well as prediction of the share of choices.
The individual response model processes the consumer measures by
"reducing" them to an underlying set of perceptual dimensions. "Abstraction"
defines homogeneous groups based on perceptions and preference. "Compaction"
describes how the reduced space performance measures are combined to produce a scalar measure of goodness for each consumer and for each choice
alternative. This goodness measure is linked to probability of choice for
the new and old alternatives. In each step, theoretical, empirical, and
statistical issues are identified and various techniques are described
for each phase.
The techniques are demonstrated based on survey data collected at MIT
to support the design of a health maintenance organization (HMO). After
discussing the issues of testing the model, the managerial design implications are shown by application to the MIT HMO case.
MODELING DECISIONS OF CHOICE AMONG FINITE ALTERNATIVES
APPLICATIONS TO MARKETING AND TO TRANSPORTATION DEMAND THEORY
by
John R. Hauser
ABSTRACT
Many of the models in marketing and in transportation demand theory address the same questions, such as, "How many people will purchase our product?"
and "How many people will ride our bus?".
This paper presents a general methodo-
logy which forces a model builder to explicitly specify his assumptions and then
allows him to easily trace out some of their implications. Most of the causal
models in both fields, particularly those which attempt to model individual
choice, are special cases of this methodology. Two examples are given:
an
existing technique (disaggregate behavioral demand models) and a new technique
(direct utility assessment).
i
ACKNOWLEDGEMENT
I wish to take this opportunity to thank Professors John D.C. Little and
Glen L. Urban for their guidance and help in formulating this methodology.
I
wish also to thank Ralph L. Keeney and Samuel E. Bodily for their suggestions
on the direct utility assessment technique and for their comments on the
methodology.
Further thanks go to Pitu Mirchandani and Gabriel Bitran for their moral
support and to Martha Yee for her excellent typing.
ii
TABLE OF CONTENTS
PAGE
SECTION I: INTRODUCTION
1
SECTION II:
2
SECTION III:
CRITERIA
5
THE BASIC METHODOLOGY
3.1
Choice Model Interacts with the Design Process
5
3.2
Observation of Consumers
7
3.3
Individual Choice Behavior
7
3.4
Group Response
11
3.5
Interaction with Design Process
11
3.6
Scope of the Methodology
13
SECTION IV:
17
TWO EXAMPLES
4.1
Disaggregate Behavioral Demand Models
17
4.2
A Cardinal Utility Theoretic Approach
20
SECTION V:
APPENDICES:
27
RESEARCH OBJECTIVES
Review of Some Previous Work
Footnotes
References
iii
Al
A10
All
LIST OF FIGURES
FIGURE
TITLE
PAGE
1
Analytic Process Interacts with Design Process
5
2
Analytic Process Consists of Observation, Individual
Choice Behavior, and Group Response
6
3
Submodels in the Methodology
8
4
Summary of the Choice Model
12
5
Samples of Perceptual Maps
14
6
The Complete Methodology and Its Interactions with
the Design Process
15
iv
SECTION I
INTRODUCTION
Many designs require the prediction of consumer response to new alternatives or to changes in existing alternatives.
Consider the following examples:
Transportation
A community considering the introduction of a new dial-a-ride system requires predictions of demand.
Health
A university medical department wishing to design its medical plans to
best serve its potential users needs to know how many people favor which plans.
Finance
A brokerage house contemplating new services, wants to know how popular
they will be.
A country offering tourist services, a company selling deodorant, an agency
offering counseling service, all share a common need; the need to know how consumers will respond to a choice among a final number of alternatives.
As can be surmised from the variety of applications, the field of choice
prediction is now new.
Many researchers in diverse disciplines have developed
models of choice behavior.
(A brief summary of some of these efforts appears
in Appendix I).
This paper endeavors to develop a general methodology which includes most
of this previous work as special cases.
Before describing the basic methodology,
let us first consider some criteria for the development of a choice prediction
theory.
-1-
SECTION II
CRITERIA
Choice Predictions must be Theoretically Sound, have Measurable Inputs,
and be Responsive to the Decision Process
The design decisions mentioned in the introduction require much more than
the prediction of who will choose what.
They require a choice prediction
methodology which interacts well with the decision process, i.e., which reveals
why consumers are reacting the way they are, indicates how to improve the consumer response, and guides creativity in the design of new alternatives.
The primary goal of this paper is to present a methodology which is as
general as possible, as theoretically sound as possible and which is very responsive to design decisions.
Specifically, the methodology will be:
General
1. Diverse - Each step in the methodology should offer a variety of techniques of varying complexity and data requirements.
In this way, the methodology
can respond to the diverse needs of decision makers and data availability.
In
addition, it can adapt to changes in requirements or in data.
2. Encompassing - The methodology should deal with choice behavior in
general and as such can be used for all the applications mentioned in the introduction.
3. Inclusive - Almost all of the previous work on choice behavior should
be covered by the methodology as special cases.
This allows the incorporation
of proven techniques and the exchange of subsets of techniques.
Theoretically Sound
4. Reflect Behavior - While not explicitly requiring microscopic models
of behavior, the methodology should be consistent with what is known about be-2-
havior, taking account of the level of aggregation and simplification required.
5. Transparent - All models require assumptions.
The methodology should
make its assumptions explicit and force the submodels to make their assumptions
explicit.
In doing so, the methodology isolates weaknesses in existing tech-
niques, and indicates where improvements need to be made.
In addition, it pre-
vents models from being used in applications which violate their assumptions.
6. Mathematically Consistent - Once the assumptions are isolated, the conclusions reached by the methodology must follow from these assumptions by the
use of consistent mathematical logic.
Useful Predictive Powers
7. Have Measurable Inputs - The inputs must be currently available or it
must be feasible to obtain them.
Judgmental inputs, if required, must be
feasible with the technical capabilities and experience of the design team.
8. Controllable - The methodology must predict response to changes which
the decision process can control.
For example, a model to be used in deter-
mining the geographic location of a transportation system may have weather as
an independent variable, but a model for determining changes in frequency of
service on an existing system should not.
9. Extendable - The methodology should not be simply descriptive but should
be normative.
It should have the capability of predicting response to new or
changed alternatives rather than simply describing how consumers respond to the
existing alternatives.
Elicit Creativity
10.
Identify Characteristics - By identifying how the consumer perceives
the alternatives, i.e., by identifying which characteristics are relevant to
the choice decision, the methodology can indicate in which directions the de-3-
sign should go.
11.
Identify Tradeoffs - It is often possible to improve one characteris-
tic at the expense of another; thus, in the design process it is necessary to
know which characteristics are most important and by how much.
Acceptable to the Design Team
12.
Understandable Process - Although some steps can be viewed as "black
boxes," the overall methodology should be understandable to non-technical members of the design team.
13.
Understandable Output - Presentation of the results of each step in
the methodology should be clear and understandable to the design team.
Every
attempt should be made to enable the design team to visualize the underlying
choice processes.
14.
Robust - If possible the methodology should have some natural ro-
bustness, i.e., it should be hard to get absurd answers.
-4-
_
-
-
__
I_
SECTION III
THE BASIC METHODOLOGY
This section presents the basic structure of the methodology.
Ongoing
research will develop this structure in more detail than is presented here,
summarize and critique existing techniques for each step in the methodology,
develop a formal theoretic structure for some steps, and introduce new techniques wherever appropriate.
In addition, a testing technique will be developed
which gives a measure of overall performance, as well as dividing that measure
among the various steps.
3.1
Choice Model Interacts with the Design Process
As was argued in Section I, it is imperative that the choice model be in-
timately tied to the design process (see Figure 1).
Design Process
I
Analytic Process
1---~·--~
I ___
I
I
Predict
Consumer
Response
Decisions on
Change
--A
,I
_
_
-----.lr-1-_ _
Figure 1:
AI
_~~~~~~~~
__..__._.__1
·L4
Analytic process interacts with design process.
-5-
_
--- -----;p-^n
i
In other words, the design team should be able to change alternatives on
many levels, e.g., specific:
fortable waiting rooms.
waiting rooms 50% larger, or general:
more com-
Conversely, the choice model should provide insight
on many levels in addition to just predicting the number of people choosing the
alternative.
For example, it should identify relevant characteristics, their
importance, and how the quality of each combine to alter choice probabilities.
It should also indicate the variation of behavior across the population.
Since, ultimately, it is consumers who are choosing among alternatives,
it is necessary to model individual behavior in order to gain insight on consumer response.
Thus, the choice model is divided into six steps.
The first
two are observational, the next three model individual choice behavior in such
a way to elicit creativity from the design team, and the sixth is an aggregation step which transforms individual behavior into overall market response
(see Figure 2).
Design Process
Figure 2:
Analytic Process
Analytic process consists of observation individual
choice behavior, and group response.
-6-
_
-1-sil 1 1_1_ II___I_·^____
LII·I--L--I__-IIIYXII-I-I
-._._ _
3.2 Observation of Consumers
Observation is modeled as a two-step process to explicitly identify potential errors.
The first part is presenting the individual with a choice. This
can be a real choice or a proxy design to elicit choice behavior which approximates what it would be in a realistic choice environment.
For example, in-
dividuals might be observed on which mode of transportation they choose from
the existing alternatives, or they might be given descriptions of new alternatives and asked to choose among them. The second part is measurement. This is
somehow identifying quantitative measures of consumers' perceptions of the alternatives, consumer characteristics, and choice. They can be cardinal, e.g.,
travel times on various modes or ordinal, e.g., a rank order on the availability
of doctors in health maintenance organizations.
These measurements are input into the next section, choice behavior.
3.3
Individual Choice Behavior
The numerical goal of this section is to transform the observed measures
into individual choice probabilities.
That is, obtain estimates for each in-
dividual of the probabilities that that individual will choose defined alternatives.
But, the more important goal is to provide insight into the individuals'
choice behavior and to elicit creativity in the design of new or changed alternatives.
To provide this insight and to approximate choice behavior, the measures
are transformed in three identifiable steps (see Figure 3).
Reduction/Abstraction - From many measures (e.g., 20), the data is reduced to a few performance measures and preference measures (e.g., 4 and 4).
The reasons for this are many fold, but mostly dealing with the inability of
the human mind to explicitly consider more than a few measures simultaneously.
-7-
I
L _
--
Present the
individual with a
w choice
-4
Measurement
2I
J1---P
- characteristics
- preferences
- choice
real vs. proxy
- perceptions
a) Observation of consumers
Reductior
Absttraction
-a
s
ci
Cij = C s(x j,xi)
X, on)
(segmentatLion)
Probability of
Choice
Compaction
I Pij
PS(aj I.i
)
~--
-1
!j
.
.
.
b) Individual choice behavior
w
. .
_~~~~~~-
market share
P.
1j
Aggregation
.-
c) Group response
Figure 3:
Submodels in the methodology
-8-
-- -- ----
-·- ------------------- II
-
4
*b
variations
k--
Thus the reduction of dimensionality and the abstraction of the relevant performance measures allows the design team to better understand which characteristics are relevant to the choice.
Furthermore, a geometric representation of
the data in fewer dimensions allows visualization of the choice process.
Of course, reduction/abstraction does represent a behavioral assumption,
the assumption that in order to make a choice, the individual somehow conceives of the alternatives along a relatively small number of dimensions.
If
the design team feels uncomfortable with this assumption, reduction/abstraction
becomes a null step and the measures from the observation section are passed
directly to the next step in the methodology.
In addition to reducing the data, a certain amount of segmentation occurs
at this step.
Because preference measures can vary significantly from individual
to individual, an attempt is made to link these variations to population characteristics and to abstract appropriate relatively homogeneous segmentations.
Such segmentations should be necessary (variation occurs), identifiable, and
relevant to the design process.
Compaction - Even with only a few dimensions to the data, the design team
needs to know the saliencies of each dimension.
In other words, how do in-
dividuals trade off the performance measures, how interdependent are the performance measures, and what are the risk characteristics of the individuals'
choice processes. Thus, a compaction function is determined and for each
individual the performance measures for each alternative and the preference
measures are compacted into a scalar measure of goodness for that alternative.
This intermediate step allows the design team to understand how the performance measures combine and thus guides the decisions they must make in the
design of alternatives.
For example: "increasing the safety of the train at
the expense of speed" or "Increasing parking outside a medical department is
-9-
-~~~
useful only if the waiting rooms are made more comfortable."
Compaction is intimately tied to the next step, probability of choice, in
that every compaction function, c(x, x), mapping the performance measures into a single number must have the property that with all other alternatives held
fixed, the same value for the scalar measure of goodness for an alternative
must yield the same probability of choice for that alternative.
In other words,
knowing the measures of goodness for all the alternatives is sufficient for
determining the choice probabilities.
Notice that a compaction function is
more general than a utility function because (1)its domain includes preference
measures as well as performance measures and (2)there is a randomness to behavior in that the individual does not always choose that alternative with
the largest compaction value.
Again, compaction is a behavioral assumption and if found unacceptable, it
becomes a null step and the performance and preference measures are transformed
directly into choice probabilities.
Probability of Choice - This last step in the choice behavior section
mathematically transforms the vector of compaction values into choice probabilities.
Reduction/abstraction and compaction provide qualitative guides to de-
sign; this last step provides probabilities that can be used to get the numerical implications of design decisions.
probabilities, i.e.,
The output of this step can be Bernoulli
(possibly non-stationary) estimates of probabilities of
selection for any choice occasion, or they can be Poisson rates, i.e., if X is
the choice rate for an alternative, then the probability that the individual
will choose that alternative in (small) time At is AAt.
Of course, inputs for
more complex models are consistent with the methodology, but techniques for
their determination will not be developed in this research.
-10-
__
I
__I__
3.4 Group Response
Aggregation - The last section of the choice model is aggregation of individual choice probabilities (refer back to Figure 2).
The design team is
ultimately concerned with total choice behavior; thus, the individual choice
probabilities must be combined to produce measures of overall consumer response.
This aggregation explicitly uses the segmentations identified in the choice
behavior section to extrapolate from the sample population to the target population.
Some segments, e.g., elderly citizens, may be handled separately, if
doing so is relevant to the design process.
The typical output of aggregation is often a single numerical estimate
of market share for each alternative, but to be fully useful, measures of
variation of share such as covariance should also be output.
In addition, by
analytically or numerically determining gradients of market share with respect
to the performance measures, sensitivity analysis can be done.
A summary of the choice model portion of the methodology appears in
Figure 4.
3.5
Interaction with Design Process
The choice model provides insight on four levels (1)reduction/abstraction
identifies the relevant performance measures and segmentations, (2)compaction
identifies tradeoffs, interactions, and risk characteristics among the performance measures, and condenses them to a scalar measure of goodness, (3)probability of choice gives numerical implications of the scalar measure of goodness, and (4)aggregation provides measures of aggregate consumer response and
variation in that response.
Each level provides information, but in order for that information to
provide useful insight to the design team, it must be presented to them in a
-11-
Analytic Process
Design Process
Present Choice
.
.
.-
-
-
.
.
Observe
Consumers
real vs. proxy
-
Changes in
A.lternatives
r~~~~~~~~~ii
_
.
Measurement
'
i-
-
-
n
Reduction/
Abstraction
-'.
A -- X, A
-.
E;
--
1_-1
Model
Individual
Choice
Behavior
Compaction
Cij = Cs(xX)-i
J.I
·
Probability of
Choice
rl
j = Ps(aj
)
Decisions on
Change
11P
Predict
Group
Response
Aggregation
I
-
Pij ------
------
,
m-;
_-
Figure 4: Summary of the Choice Model
-12-
I
-·-
·----
msj, ms
,,,__
form they feel comfortable with and can mentally manipulate.
For example,
in the reduction/abstraction step, a map (Green and Wind [27], Urban [86]) of each alternative in "average" perceptual space might be presented, (Figure 5a), or
perhaps a scatter diagram of individuals' perceptions of the alternatives in
the same perceptual space, (Figure 5b).
These presentations can be in terms
of numbers (means and variances), maps, sensitivity curves or whatever the design team feels comfortable with.
Just as presentations can occur from many steps in the methodology, changes
in design decisions can be input for testing at many steps.
For example, im-
plications of a very specific change like doubling the frequency of bus service
on an existing route can be tested by changing an input to the reduction/abstraction step, or a general change like improving comfort can be tested by
changing an input directly into the compaction step.
Similarly, an entirely
new concept might require facing a sample population with a new set of alternatives, or might simply require taking a few additional measurements.
Pro-
motion and/or advertising is tested by determining their implications as changes
in the preference measures.
Figure 6 summarizes the methodology.
3.6 Scope of the Methodology
By its nature, developing a general methodology is an ambitious task, but
its generality does not imply a once and done procedure.
Its use does not en-
tail cycling once through the analytic process, determining the market share
and then stopping.
In fact, the same methodology, with different submodels,
can be used during many different phases in the design process.
In the search for new alternatives or changes in existing alternatives,
reduction/abstraction is particularly useful because it identifies which at-13-
i
LPerformance
, subway
* automobile
comfort and
convenience
bus
o
a) "average" perceptual space
A performance
5
S
b
..
,
-
g
-Fr
comfort and
convenience
b
b) scatter diagram in perceptual space
Figure 5:
Samples of perceptual maps
-14-
Analytic Process
Design Process
i-
i
m
i]
Observe
Consumers
A Present Choice
I
real vs. proxy
#-
Changes in
Alternatives
A-
_
-I-
-9
.
Measurement
I
estimate change
in
_
___
X or A
_
i
Reduction/
-1
II
Abstraction
.--1_7_____
X, A
_.
N
.~i~
Compaction
/ C.j - CS(xj,
Presentation
Model
Individual
Choice
Behavior
)
Probability of
Choice
Pij = Ps(aj Ic)
Decisions on
Change
_
_~~~~~~~~~~~~~~~~~~~t
Predict
Group
Response
,&
Aggregation
U(sales, cost,
etc.)
i---
Pijj- msj, msj
--
_
.
J
__
Figure 6: The complete methodology and its interactions with the
design process.
-15-
tributes are most important and which ones combine to form performance measures.
In addition, first cut compaction functions provide insight on tradeoffs, risk,
and interaction among performance measures.
All of which are useful in search.
In screening, dominance can be checked, i.e., if each of the performance
measures of one alternative are dominated by another.
Also, early prediction
of market share and explicit modeling of the uncertainty in its estimation allows early pruning of a large set of alternatives.
In some circumstances, such as the microscopic design of a bus network,
the methodology is used directly to help determine the initial design.
In
other circumstances, such as the design of health maintenance organizations,
it does not replace test markets, but instead helps in the design of the alternative(s) which are to go to test market.
During test market, (or laboratory simulation in the case of frequently
purchased consumer products) the methodology is used with more complex submodels
to improve understanding of consumer behavior, and to extrapolate the entire
consumer response from test market results.
Finally, during implementation, the methodology is used to monitor consumer response and to suggest improvements in the alternative, if necessary.
-16-
SECTION IV
TWO EXAMPLES
Probably the best way to explain a methodology is with examples.
This
section presents two methods to predict the market shares of competing transportation alternatives. The first consists entirely of existing techniques,
The second con-
many similar to those used by Quarmby, Ben-Akiva, and others.
sists mostly of new techniques currently being developed at M.I.T.
4.1
Example 1:
Disaggregate Behavioral Demand Models
Suppose for simplicity that total demand is known and we are only predicting market share for two alternatives, bus and automobile.
Observation
Presenting the Individual with a Choice - The observation is made in a
real choice environment, i.e., the individual is questioned on the choice he
most recently made when faced with the existing alternatives.
This has the
advantage that repeat probabilities rather than trial probabilities are determined, but the disadvantage that if entirely new options are introduced, the
values of their attributes may be well outside the predictive range of the
model.
Measurement - The choice behavior, the values of the attributes on existing
modes, and demographic data are measured via questionnaire.
Thus, some noise
is introduced because of potential misunderstanding of the questionnaire. An
attempt is made to relate perceived time to measured time1 because although an
individual makes a choice based on perceived time, prediction will be based on
measured time.
-17-
Choice Behavior
Reduction/Abstraction - Although some is done by combinatorial experiments and some by dropping variables which are not statistically significant,
this step is based primarily on the model builder's professional judgment and
experience.
For example, travel time, wait time, cost, access time, a dummy
variable for mode, and dummy variables for income are a few of the candidates
for explanatory variables.
Segmentations might be by trip purpose, i.e.,
work trips vs. non-work trips.
Compaction - The compaction function is a linear "disutility" function,
i.e., this scalar measure of goodness for each mode is calculated via a weighted
sum of the variables.
Ting [81] shows that under the assumption that every pair of attributes
is preferentially independent2 of all others, an individual's cardinal utility
function can be written as a function of an additive value function.
words, there exist continuously differentiable functions V,gl,g2
u(x) = V[g2(x
2
) + ...
.gm
In other
such that
+ gm(xm)].
Unfortunately, the theory gives no easy way to specify these functions
and their form can be quite complex.
The use of a linear "disutility" function
assumes that these value functions can be approximated by linear transformations of the variables.
In addition to the assumptions of preferential independence and linear
value functions, an assumption is made that the "disutility" function and the
values of the weights are identical for all individuals within a segment.
The
values of these weights are determined in the next step, probability of choice.
Probability of Choice - In this step, a conditional probability law is
determined which calculates the probability that an individual will choose
-18-
one mode based on the scalar measures of goodness for each mode.
The following
assumptions are made:
1. Individuals are independent.
2. An individual always chooses that mode which maximizes his true
scalar measure of goodness (negative "disutility"), but
3. The true scalar measure of goodness is equal to our calculated
scalar measure of goodness plus a random disturbance term, i.e.,
dtrue
dcalculated +
4. The disturbance terms are independent and identically distributed
with Weibull densities, i.e.,
P (co) = e
exp(-e
)
These assumptions imply that the probability of choosing mode a1 is determined by the logit model, i.e.,
-d
-d
P(a
1 I d1 , d2 ) =
e
2
+e
The values of the weights are determined either by maximum likelihood or
by linear regression, (Note that ln(l-p
) = d2 - dl where p = p(al
d1, d2).)
and variables which are not statistically significant are dropped.
Aggregation
Aggregation is a difficult step in transportation demand prediction since
model splits (market shares) are often required for each zonal interchange [99]
and because the values of the explanatory variables can vary considerably from
individual to individual.
calculations required.
Thus, the real problem is the tremendous number of
Various techniques exist for simplifying these calcu-
lations but basically grand means are calculated from individual probabilities.
-19-
In other words, if the explanatory variables are distributed across the
population segment with joint probability distribution p(x
1,
2)
then the
predicted market share of alternative 1 for that segment is:
ms1
=
/P[a1 I dl(x 1)
d 2( 2 )] P(1' x2) dxl d
2
Equation 4.1
Interaction with Decision Process
The procedure just described is primarily a prediction procedure rather
than a design process.
Its strength is in predicting market share for already
defined alternatives.
It does provide some design insight because the weights
for the explanatory variable are indicators of the strength of the effect that
variable has.
Care must be taken in interpretation because of the potential
multicolinearity among the variables.
In addition, if the assumption of homogeneity of weights within a segment
is valid, elasticities can be easily calculated.
4.2
Example 2:
A Cardinal Utility Theoretic Approach
Again, for simplicity, assume that these are only two alternatives, bus
and automobile.
Observation
Presenting the Individual with a Choice - This technique makes use of both
behavioral and preference data.
Thus, the individual is faced with a real
choice among existing alternatives and an artificial choice among attributes
of potential alternatives.
This enables the model to predict choice for al-
ternatives with values well outside the range of those available on existing
modes.
Measurement - The choice behavior, the perceived values of the attributes on existing modes, demographic data, and the preferences toward the at-20-
tributes are all determined by questionnaire. The greatest potential for error occurs in this step because of the difficulty a consumer might have in
understanding preference questions.
The three types of preference questions asked will be similar to the following:
Tradeoff Questions
1. Suppose the mode of transportation you are using costs $1.00. Suppose you can expect a waiting time of 10 minutes and a travel time
of 20 minutes.
A more reliable mode is offered which also costs $1.00. This mode
guarantees only a 5 minute waiting time. What is the maximum travel
time you would accept and still prefer this more reliable mode?
Risk Questions
2. Suppose the mode of transportation you are using costs $1.00. Suppose
you are not sure of the waiting time, in fact, it is as if someone
flipped a coin: heads meant you had to wait 5 minutes, tails meant
25 minutes. In other words, an average time of 1/2(5 + 25) = 15 minutes.
A more reliable mode is offered which also costs $1.00. This mode
can guarantee a fixed wait time. What is the maximum guaranteed wait
time you would accept and still prefer this more reliable mode?
Independence Questions
3. If both the existing and the new reliable mode cost only $.50 would
your answers to questions 1 and 2 change? If so, what would they now
be?
The independence questions determine the mathematical form of a utility
function while the risk and tradeoff questions determine the preference parameters of each individual utility function.
(More on this in the discussion
of compaction.)
Choice Behavior
Reduction/Abstraction - This step has two phases, exploratory and actual.
The exploratory step can be performed on existing data.
-21-
It makes use of statis-
tical techniques such as factor analysis and Automatic Interaction Detection
[98] as guides to professional judgment in reducing the set of potential
explanatory variables to a manageable number.
The actual reduction occurs via in depth interviews with decision makers,
other analysts (because they have thought extensively about how consumers
behave), and a relatively small number of consumers.
By iteratively asking
tradeoff, risk, and independence questions, it is possible to identify those
attributes or combinations of attributes which can be classified as performance measures, i.e., as necessary and sufficient for the decision process.
The output of this reduction is a set of potential compaction functions.
Each compaction function is an identified set of performance measures and a
functional form that that individual's utility function takes.
For example,
when this author's utility function was assessed with the aid of an interactive computer program developed by Sicherman [ 97], it was found that for
Dial-a-Ride trips the following form was a good approximation.
Performance Measures:
X 1 = travel time
x 2 = wait time
x3 = cost
Preference Measures:
c1, c2, c3 = risk aversion coefficients
k1, k2, k3, k12, K12, K = tradeoff coefficients
Independence Conditions:
1. (X1, x2) aggregatable.
-22-
2. aggregation function multiplicative
3. after aggregation, utility function multiplicative
Thus if u(x
1 , x2, x3) = utility function
w(x1, x 2) = aggregatQr function
then
1 + Ku = [1 + Kk3 v3 (x3)] [1 + Kk12w(x1, x2)]
1 + K12w
[1 + K12 klVl(x 1)] [1 + K12 k2 v2 (x2 )]
-c .x.
vi(x)
=
1 - b. + be
1
(bi normalization constants)
Compaction - Cardinal utility theory (based on the von Neumann-Morgenstern
axioms [ 90]) is normally a prescriptive technique.
In other words, it is used
to help people make decisions rather than predict how they will make them.
Thus, we might expect that everyone will not always choose that alternative
which maximizes expected utility. One of the goals of the current research effort is to develop axioms, similar to the von Neumann-Morgenstern axioms,
which apply to compaction functions, i.e., which allow randomness in behavior.
If these axioms are acceptable to the model builder, then the expected values
of an individual's scalar measures of goodness for each alternative are sufficient to predict choice.
Based on the axiomatic theory of compaction, the compaction step in this
technique consists of two phases, segmentation and distribution determination.
The final phase, segmentation, identifies segments of the population
with homogeneous compaction functions. That is, it identifies which compaction functions (as determined by reduction/abstraction) apply to which popu-23-
lation segments,
Computer packages such as AID are used to group individuals
based on their answers to the risk and independence questions.
The second phase, distribution determination, uses estimates of the performance measures for the alternatives under test to determine (the joint
probability distribution of) the scalar measures of goodness for each alternative, for each individual in each segment. Appropriately normalized, these
are used in a Bayesian choice probability model.
Probability of Choice - In this step, a conditional probability law is
determined which calculates the probability that an individual will choose
one mode based on the scalar measures of goodness for each mode.
The following
assumptions are made:
1. individuals are independent.
2. an individual does not necessarily choose that mode which maximizes
his scalar measure of goodness, but
3. two (or more) individuals from the same population segment with the
same set of scalar measure of goodness have the same choice probabilities.
These assumptions allow the use of Bayes Theorem to determine choice probabilities.
First, rank order the scalar measures of goodness.
Then let a1 be
the event that an individual chooses the mode which maximizes his scalar
measure of goodness.
p(c
1, c2
We can observe easily the posterior distributions
al) and p(c
1, c2
choosing a1 and a2.
a2) and also the total percentages, n1 and n 2,
Bayes Theorem then gives:
(al cl' C2)
P(C 1, c 2 1 a1 )
l(a
n 1p(c
,
c
1
a)
+ n2P(c
1
2
1, c2 I a 2)
(Note that if only c = c 1, -c2 matters, and if p(c
a1) and p(c a 2) are
normal with common variance, then the logit model results [Quarmby ( 66)1.)
-24-
A word about normalization.
Since both utility functions and compaction
functions are unique only up to positive linear transformations, the probability model can be sensitive to the choice of normalization. A number of
heuristics are being developed in an attempt to guide engineering judgment in
the choice of normalization.
Aggregation
The same difficulties in computation that occurred in disaggregate behavior
models occur in this technique. Because the entire methodology is normative,
the output of aggregation should include estimates of variation in market
share as well as mean values.
The question that naturally arises is:
"Can
this be obtained without tremendous additional computational effort?".
Since individuals are assumed independent, the Central Limit Theorem can
be used to determine the entire joint distribution of aggregate market shares
with little additional computation, i.e., the aggregate market shares are
jointly distributed as multivariate normal random variables with means given
by equation (4.1) (same as behavioral models) and covariances given by:
cov(msl, ms2 ) = -
fp[al I c 1 (xl), c 2(x2)] P[a 2 I Cl(X 1), c2 (x2 )].
p(X
1 ,
2)dxldx2
var(msl) = similar
Interaction with Decision Process
The procedure just described is primarily a normative procedure and is
designed to interact will with the design process.
In reduction/abstraction,
the important attributes are determined and in compaction the interactions,
tradeoffs, and risk qualities of these performance measures are identified.
For example, in the utility function on page 23, kl/k 2 measures the rela-25-
tive importance of travel time to wait time and K12 measures how strongly they
interact.
The risk aversion coefficients measure how important risk is in the
choice decision.
In probability of choice, the design team can examine the
results of design decisions directly in terms of compaction functions or in
terms of choice probabilities.
Finally, in aggregation, measures of variance
as well as means of market share can be determined.
Notice that direct assess-
ment of compaction functions avoids multicolinearity and that elasticities, if
required, can be numerically determined.
-26-
SECTION V
RESEARCH OBJECTIVES
In addition to developing the methodology in more detail, this research
will summarize and critique one or more existing techniques for each step in
the methodology.
New techniques and a formal structure will be developed as
follows.
Measurement - An examination of utility assessment via questionnaire will
be made and a few representative questions developed.
If possible, these will
be pretested for style on a small population.
Reduction/Abstraction - The issues involved in reduction will be identified through a series of rigorous definitions.
The existing techniques of
factor analysis, of automatic interaction detection, and of choosing features
with maximum information content will be examined in relation to the methodology.
Examples, based on a questionnaire for health maintenance organizations, will
be presented.
In addition, the technique of in depth utility assessment will
be developed.
Compaction - A rigorous development of the theory of compaction functions
based on axioms similar to the von Neumann-Morgenstern utility axioms will be
developed.
This will enable the utility theoretic results of Fishburn, Keeney,
Ting [20,21,23,40,41,81,90] and others to be directly applied to compaction
functions.
Independence from irrelevant alternatives and from irrelevant parti-
tions will be examined.
The mathematics for assessing compaction functions by
direct questionnaire will be developed.
Probability of Choice - Existing techniques to transform scalar measures
of goodness into probabilities, such as distance in perceptual space and
utility perturbation, will be summarized and critiqued.
-27-
A general Bayesian
technique will be developed with examples given from deodorant and from health
maintenance organization data.
Also, the problem of interpersonal comparisons
of utility or scalar measures of goodness will be discussed and some heuristics
presented to partially circumvent these problems.
Aggregation - The general mathematical equations will be developed and
the Central Limit Theorem approximation will be introduced.
A discussion of
numerical problems will be made, but no numerical examples are planned at this
time.
Testing - A testing technique will be developed based on honest reward
functions, maximum likelihood and information theory. This test tries to
measure how "good" probability measures really are.
Communication with Design Process - Various techniques such as pictorial
representations will be examined for presentation of the outputs to the design
team.
-28-
APPENDIX
REVIEW OF SOME PREVIOUS WORK
Most (but not all) of the work in choice modeling has been done in three
fields:
transportation demand modeling, marketing research, and mathematical
psychology. The purpose of this appendix is not to summarize all the work in
each field (for that could take many volumes) but to give an indication of the
types of issues involved and of various approaches taken.
Al.1
Transportation
Urban Transportation Model System - Perhaps the most classical approach to
demand modeling is the "Urban Transportation Model System" (UTMS), which has been
applied in over 200 cities in the U.S. over the last 17 years.
(Manheim [55])
This process models consumer choice as sequential, i.e. the consumer first
decides to travel (trip generation), then chooses a destination (trip distribution), picks a modeoftravel (modal split) and finally, if multiple paths exist
from origin to destination, chooses a path of travel (assignment).
Feedback
loops are often incorporated to obtain stable estimates (because the volume of
travel effects the quality of service which in turn effects volume) but still
the process is essentially sequential.
Most of the submodels used in the UTMS
have been aggregate in the sense that individual choice has not been explicitly
modeled.
For example; Trip
generation:
Land use, automobile ownership, etc. are
extrapolated from current trends and total trips determined by multiple linear
regression (Douglas and Lewis [17], Fleet and Robertson [24], Martin, Memmott
and Bone [57]).
Trip distribution:
The fraction of trips to a given destination
zone is a function of the "size" of the zone, the average travel time to that
-Al-
zone, and other measures of the level of service. Arbitrary functional forms,
such as the gravity model (U.S. Dept. of Transportation [87]) are used as are
aggregate probabilistic models such as the opportunities model (Ruiter [70],
Witheford [93], Hauser [31]).
Modal split:
The market share each mode captures
is a function of the average level of service on that mode. Techniques to do
this are diversion curves (Traffic Research Corp. [82]) and other formulas
(Wynn [94]).
Assignment:
(Disaggregate approaches to modal split will be discussed later.)
this is done mostly by mathematical programming such as minimum
path assignment techniques.
Direct Demand Models - The UTMS models are sequential, but many analysts feel
decisions to travel are simultaneous in nature. That is,mode, destination,
path, and frequency are highly interdependent decisions.
The first attempts to
incorporate this posulated interdependence were (aggregate) direct demand models
(McLynn [53], Quandt and Baumol [64], SARC [79]).
Direct demand models are
essentially econometric models where the analyst postulates a function form
such as:
dl
ijk
where
a ca2 (P )a 3
ijk ijk(ipi)
dij k = number of trips between city pair
i -+ j on mode k
cijk = cost of travel . . .
tij
k
= travel time . . .
Pi = population of city i
S, a1, a2 , a3 = constants
In this example a logarithmic transformation makes the equation linear in the
parameters which are then determined by regression.
These models have the advantage of treating choice as simultaneous, but
-A2-
have many disadvantages.
Among these are that (1)the functional forms,
although motivated by behavioral reasoning, still have a certain arbitrariness
to them, (2)multicollinearity among the variable often makes interpretation
difficult, (3)they are calibrated on existing alternatives and thus extrapolation to new alternatives or radically different service is difficult, and
(4)because aggregate data is used, much of the detailed individual choice is
not modeled.
Disaggregate Models - A more recent development in demand models have been
attempts to model individual choice.
Quarmby [66], Stophes [77]).
For example
(Lave [45], McFadden [52],
Most of the applications have been in predicting
modal choice although some, most notably Ben-Akiva [6], have attempted to
simultaneously predict mode, destination, and frequency.
One of these models is described in more detail in example 1, section IV
but basically they attempt to predict individual choice probabilities based on
quality of service (and demographic) measures for each mode.
A1.2
Marketing Models
In marketing the problem is to predict how consumers will respond to new
and/or mature products.
Some of the important issues in these models are
(1)normative vs. descriptive, (2)the degree of aggregation, (3)the degree of
behavioral modeling, (4)qualitative vs. quantitative outputs, (5)type of
input, and (6)extent of application.
Normative vs. Descriptive:
behave the way they do.
A descriptive model strives to explain why consumers
Whereas a normative model endeavors to predict consumer
response to management options.
Because of the complexity of consumer behavior, descriptive models tend to
-A3-
have more detail and are formulated with the goals of learning about consumers.
Some examples of descriptive models are Herniter's entropy model [33,35], Bass'
theory of stochastic preference [5], and Butler's "Hendrodynamics" [10] which
model consumer behavioras inherently stochastic and choice probabilities are
determined by optimizing some measure of randomness subject to observable
constraints such as market share.
Normative models also attempt to describe consumer behavior but with the
goal of predicting how management decisions affect the behavior.
For example
Little's BRANDAID [48] relates price, advertising, promotions, and other variables
to aggregate consumer response in a way that allows relatively simple evaluation
of decision strategies.
Calibration steps include judgment, analysis of
historical data, tracking, field measurement, and adaptive control.
In other
words consistent use is made at all available information to uncover problems,
focus managerial effort, enhance insight on market structure, and guide the
decision making process.
Degree of aggregation - Ultimately most managerial decisions will be made
on aggregate response, but models differ in their technique of obtaining this
response.
Some deal directly with the individual, modeling each consumer's
decision, while others use "average" inputs.
For example Burger's new product forecasting system [9] observes individual
consumers and estimates probability of purchase from a multiple linear regression model.
Prediction is accomplished by first projecting the independent
variables (preference, distribution, awareness, intent, and price) then predicting individual probabilities from the regression model.
An example of an
aggregate model is Montgomery and Silk's communication expenditures model for
ethical drugs [60].
In this model, market share is considered a function of
(distributed lags of) expenditures on communications mix variables such as
-A4-
journal advertising, direct mail advertising, and samples and literature.
Degree of behavioral modeling - More detail in the description of consumer
behavior means greater the potential for predictability, but greater detail also
means greater expense in the use of the model, greater difficulty in obtaining
parameter estimates, and occasionally less of an ability to isolate essential
consumer dynamics.
Depending upon its intended purpose different models have
varying degrees of behavioral modeling.
For example, the microsimulation models of Amstutz [2] and Herniter and
Cook [36] simulate the buying decisions of each and every consumer.
have a large number of parameters which,
Such models
when set appropriately, enable the
model to replicate behavior, but they are difficult and expensive to use for
testing new strategies.
A model with a different degree of behavioral modeling is repeated in
Claycamp and Liddy [14].
This model, which identifies causality but does not
simulate individual behavior, is a multiple equation regression model which
tries to predict percent of initial purchase for a product based on independent
variables such as "coverage of consumer promotions adjusted for type and value
of offer."
A model which is somewhere between the extremes of behavioral modeling is
Farley and Ring's [19] empirical test of the Howard-Sheth model [39],
In this
formulation a "conceptual model of buyers behavior" is cast in a regression
format.
In other words, this model identifies causalities in the consumer's
buying decisions and approximates the relationships by linear equations.
Qualitative vs. quantitative outputs - The primary purpose of some models
is to make a GO/NO GO decision and as such the model is required to produce
-A5-
numerical estimates of market share. Other models are primarily diagnostic and
as such the model is required to produce qualitative indicators.
For example in Stefflre's [76] studies of brand similarities and in Urban's
PERCEPTOR [86] multi-dimensional scaling techniques are used to determine which
attributes of a product are important in buying behavior. (See Green and Wind [27]
or Rummel [71] for a discussion of multi-dimensional scaling techniques.)
Urban goes on from there to develop quantitative estimates of market share
based upon the distance in perceptual space between a product and an ideal point.
Types of measurement - The types of measurement used are highly dependent
both on a model and its application.
For example Ahl [1] and Massey [58] use
consumer panels (diary of purchases) to forecast national demand data collected
on existing products to develop dimensions of need and consumer semantics.
Urban and Silk [96], and Burger [9], use a simulated store to test new products
and new product concepts. Herniter [35], Butlers [10], and Bass [5] use primarily,but not exclusively,market share data.
Etc.
Extent of application - Montgomery and Urban [62] categorize a new product
planning system as a four-stage process; search, screen, analyze, and implement.
Search is the creative generation of new ideas, screening is the narrowing
down to a few promising ones.
The ideas are then analyzed for their potential
and the best set is implemented first in lab simulation or test market, and
then in national rollout. Different models attack different phases of this
process.
For example, Urban's PERCEPTOR [86] and Stefflre's similarities [76]
are used primarily to structure and assist in idea generation and to provide
an initial screening mechanism.
Urban's SPRINTER MOD III [85] and Ahl's
forecasting systems [1] are designed to analyze test market results.
Finally
Little's BRANDAID [48] is used to aid managerial decisions dealing with mature
products.
-A6-
Other models and issues - The previous list of issues is not meant to be
exhaustive nor are all the important models mentioned.
For example there are
sales force allocation models of Davis and Farley [15], Lodish [50], and
Montgomery, Silk, and Zaragoza [61], or the market share theorem of Little and
Bell [49], or the semi-markov descriptive model of Herniter [34], or diffusion
of innovation (Rogers and Stanfield [69], Utterbach [88]) and new products (Bass[41]).
A1.3
Mathematical Psychology
Most of the emphasis in mathematical psychology on choice behavior is on
descriptive rather than normative models.
For example, much work has been done
on the measurement of attitudes.
In 1927 Thurstone [80] postulated his "law of comparative judgment" which
states that individuals respond to stimuli according to a discriminal process.
In other words, for a set of stimuli there exists some psychological continuum
(scale) and the scale value that an individual perceives for each stimuli is
normally distributed about some mean. An individual then discriminates between
two stimuli (prefers one to another) based on the maximum perceived scale value.
Thurstone then proposed a measurement technique, the method of paired comparisons,
to directly estimate the mean scale values for each stimuli.
Many other scaling
techniques have since been studied such as Thurstone's method of equally
appearing intervals and his method of successive intervals, Guttman's Scalegram
analysis, and Coombs unfolding analysis. An excellent summary of these techniques
appears in Green [36].
In addition to one-dimensional scaling techniques much work has been done
on multidimensional scaling techniques.
For example, factor analysis
(Rummel [71]) is a statistical technique which obtains a lower dimensional
-A7-
choice theory
In the context of
of a data set of large dimensionality.
representation
this identifies the dimensions which affect choice and describe
them as parsimoniously as possible.
Other multi-dimensional scaling techniques
are the unfolding analyses which develop perceptual maps (metric data) from
nonmetric rank order input. These and other multi-dimensional techniques are
described in Green and Carmone [28] and Green and Wind [27].
Another approach to the description of individual choice behavior is to
begin with a fundamental set of choice axioms and derive models using deductive
reasoning.
Thurstone's law of comparative judgment is once such axiom and his
model of behavior is representative of a class of models which have come to be
known as "random utility models."
One such random utility
(McFadden [52])
model, the logit model, isdescribed in section IV, example 1 of this paper.
Luce [51] begins with a choice axiom which (essentially) states that the
probability of choosing some alternative, say aj from a choice set A is equal
to the probability of choosing aj from B
A times the probability of choosing
B from A. I.e.
ProbjIchoose aj from A
= Prob
Prob Ichoose B from A}
*
lchoose aj from B
This axiom is realistic if the choices,aj, are distinct independent choices
and Luce derives many useful results from his axiom.
A generalization of Luce's
axiom is the asiom of simple scalability which says that there exists a scale,
u(x), which assigns to each alternative a real number and that each choice
probability is a monotone function of that scale.
(Tversky [83]).
These axioms, although intuitively pleasing, lead to counterintuitive
results if an alternative is added to the choice set which is very similar to
some but not all of the alternatives already in the choice set.
-A8-
For example,
suppose that given the choice between driving and walking along path 1 you will
choose driving 60% of the time.
Suppose further that there exists an alterna-
tive path, path 2 and that you are indifferent between path 1 and path 2.
Luce's axiom then implies that given the choice between driving or walking path 1
vs. walking path 2, you would drive only 43% of the time, [.43 - .6/(.6 + .4 + .4)]
whereas a rational hierarchical model would imply driving 60% of the time.
Tversky [83] proposes an alternative formulation called "elimination-byaspects."
The motivation behind this model is that individuals first select an
aspect (e.g. automatic transmission in the choice of a new car) and eliminate
all alternatives that do not satisfy that criteria. This model is closely
related to lexicographic rules (Fishburn [22]) but differs because the order
in which aspects are selected is random.
Tversky then goes on to show that
Luce's formulation is a special case of elimination-by-aspects.
A1.4 Other Choice Models
There are many other choice models in the literature, for example "migration in social demography, voting behavior in political science, and mortality
in bioassay." 3
A survey of these and other choice models is contained in
McFadden [52].
One which is not mentioned in McFadden is Boyle's [8] use of
information theoretic pattern recognition algorithms to predict whether an
individual will default on a consumer loan.
Finally there is the prescriptive utility theory of decision analysis
which is used to aid in decisions rather than predict decisions.
See for
example Raiffa [67], Keeney, [40.41,42], Ting [81], and Fishburn [20,21,23].
-A9-
FOOTNOTES
1. Calculated from minimum paths along mathematical networks representing
the actual transportation system.
X = the set of attributes
2. Let
Y,Z = partitions of this set i.e.
Xl'
2
X; yl12
X= Y x Z
Y; z
z EZ
Write
x > x2
to mean x1 preferred tox
2
Then Y is preferentially independent of Z if for some z°
(Y1z') > (y2' Z )
implies that
(-1
for all z
Z ) > (Y2' Z )
Z,
Z.
3. McFadden, D. "Quanta Choice Analysis: A Survey," NSF-NBER Conference
on Decision Roles and Uncertainty, March 1974.
-AlO-
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